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254 APPENDIX TO CHAPTER IX

continuity to U x [0,1], By (Ap.2.5), we are thus reduced to proving the
theorem for the mapping /: ( -> f^rt. It is clear that for n — 0, / is inessen-
tial. Suppose n ^ 0, and let us prove by contradiction that / cannot be
inessential. Otherwise, there would exist a nonconstant continuous mapping
A of U into R such that £" = e^ih(O in U. As /z(U) is a compact (3.17.9) and
connected ((9.5.8) and (3.19.7)) subset of R, /z(U) is a compact interval
[a, b] with a<b (3.19.1). Let C₀eU such that A(£₀) = a. We therefore
have £3 = e^ia\ there is a neighborhood Y of Co in U such that the oscillation
(Section 3.14) of h in V be < n; on the other hand, (9.5J) applied to the
interval ]a — nfn, a + 7c/n[, proves that there exists a point f e V such that
f " ₌ _e'<*-«>₉ where e > 0 is sufficiently small. By (9.5.5), /z(C) - (a - a) is a
multiple of 27t, and the choice of V implies that this multiple can only be 0
as soon as e < n; but this contradicts the definition of a.

(Ap.2.9) The identity mapping £-+£of\J onto itself is essential.

3. CUTS OF THE PLANE

In a metric space E, we say a subset A of E separates two points x, y
of E — A if the connected components (Section 3.19) of x and y in E — A are
distinct. We say that A cuts E (or is a cut of E) if E — A is not connected.

For any two points a, b of C such that a ^ b, let s_a> _b(z) be the function
z -* (z — d)\(z — b), defined in C — {b}; it is readily verified that s_ttj _b is a
homeomorphism of C — {b} onto C — {!}.

(Ap.3.1) (Eilenberg's criterion) Let H be a compact subset of C; in
order that H separate two distinct points a,bofC~H,a necessary and suf-
ficient condition is that the mapping z -> s_&t _b(z)/\s_ai _b(z)\ 0/H into U be essential.

(a) Sufficiency. Suppose a and b are in the same connected component
A of C — H. As C - H is open in C and C is locally connected ((3.19.1) and
(3.20.16)), A is open in C (3.19.5). By (9.7.2) there is a path t->y(t) in A,
defined in I = [0, 1], such that y(0) = a, y(l) = b. As y(t) $ H for any value
of t, the mapping (z, t)-*f(z, f) =* s_a>m(z)l\s_a>y(t)(z)\ is continuous in
H x I, and f(z, 0) = 1, /(z, 1) = s_a _b(z)/\s_a _b(z)\\ the result follows from
(Ap.2.5).

(b) Necessity. Let A be the connected component of C — H which
contains a; A is open in C and all its frontier points are in H (they cannot



	254 APPENDIX TO CHAPTER IX continuity to U x [0,1], By (Ap.2.5), we are thus reduced to proving the theorem for the mapping /: ( -> f^rt. It is clear that for n — 0, / is inessen- tial. Suppose n ^ 0, and let us prove by contradiction that / cannot be inessential. Otherwise, there would exist a nonconstant continuous mapping A of U into R such that £" = e^ih(O in U. As /z(U) is a compact (3.17.9) and connected ((9.5.8) and (3.19.7)) subset of R, /z(U) is a compact interval [a, b] with a<b (3.19.1). Let C₀eU such that A(£₀) = a. We therefore have £3 = e^ia\ there is a neighborhood Y of Co in U such that the oscillation (Section 3.14) of h in V be < n; on the other hand, (9.5J) applied to the interval ]a — nfn, a + 7c/n[, proves that there exists a point f e V such that f " ₌ _e'<-«>₉ where e > 0 is sufficiently small. By (9.5.5), /z(C) - (a -* a) is a multiple of 27t, and the choice of V implies that this multiple can only be 0 as soon as e < n; but this contradicts the definition of a.

	(Ap.2.9) The identity mapping £-+£of\J onto itself is essential.

	3. CUTS OF THE PLANE In a metric space E, we say a subset A of E separates two points x, y of E — A if the connected components (Section 3.19) of x and y in E — A are distinct. We say that A cuts E (or is a cut of E) if E — A is not connected. For any two points a, b of C such that a ^ b, let s_a> _b(z) be the function z -* (z — d)\(z — b), defined in C — {b}; it is readily verified that s_ttj _b is a homeomorphism of C — {b} onto C — {!}.

	(Ap.3.1) (Eilenberg's criterion) Let H be a compact subset of C; in order that H separate two distinct points a,bofC~H,a necessary and suf- ficient condition is that the mapping z -> s_&t _b(z)/\s_ai _b(z)\ 0/H into U be essential. (a) Sufficiency. Suppose a and b are in the same connected component A of C — H. As C - H is open in C and C is locally connected ((3.19.1) and (3.20.16)), A is open in C (3.19.5). By (9.7.2) there is a path t->y(t) in A, defined in I = [0, 1], such that y(0) = a, y(l) = b. As y(t) $ H for any value of t, the mapping (z, t)-f(z, f) =* s_a>m(z)l\s_a>y(t)(z)\* is continuous in H x I, and f(z, 0) = 1, /(z, 1) = s_a _b(z)/\s_a _b(z)\\ the result follows from (Ap.2.5). (b) Necessity. Let A be the connected component of C — H which contains a; A is open in C and all its frontier points are in H (they cannot